Mixture and Alligation is a part of the Quantitative Aptitude syllabus, and questions from this topic are asked in Bank, SSC, and other competitive exams. These questions can be tricky and time-consuming if solved using basic formulas. However, by applying the alligation rule and various short tricks, they can be solved quickly.
In the prelims exam, usually 1 to 2 questions are asked, whereas in the mains exam, 4–5 questions, including DI based on this concept, may be included. In this blog, we will explain the detailed concept of mixture and alligation along with short tricks, key concepts, and important notes.
What Is Mixture and Alligation in Quantitative Aptitude?
Mixture and Allegation is a mathematical method used to determine the ratio in which two or more ingredients or quantities at different prices or concentrations are mixed to obtain a mixture with a desired average value.
This topic is common in quantitative aptitude because it tests both logical reasoning and arithmetic speed.
Why It Appears in Exams:
- Frequently used in questions involving milk-water, acid-base, or cost-price mixing problems.
- Tests the candidate’s ability to visualize and balance quantities logically.
Skills Required:
- Ratio and proportion understanding
- Basic arithmetic
- Logical visualization
- Diagrammatic reasoning (for the allegation rule)
Why is Mixture and Alligation an Important Topic in Competitive Exams?
Every competitive exam includes one to two questions from this topic. The details of the number of questions being asked from this topic are as follows:
| Exam | No. of Questions | Difficulty |
| SSC CGL / CHSL | 1–2 | Easy |
| IBPS PO / SBI PO | 1–2 | Moderate |
| RRB NTPC / Group D | 1 | Easy |
| State PSC / Police | 1–2 | Moderate |
Terms Used in Mixture and Alligation Questions
The terms that are most commonly used in Mixture and Alligation type questions are as follows:
| Term | Details |
| Mixture | A combination of two or more components (e.g., milk and water, acid and base). |
| Mean Price / Mean Value (M) | The average value or price of the mixture. Formula: (Q₁×V₁ + Q₂×V₂) ÷ (Q₁ + Q₂) |
| Allegation Rule | Shortcut method to find the ratio in which two or more ingredients are mixed. |
| Alligation Diagram | A visual tool to apply: (D1 – M):(M – D2), where D1 and D2 are individual values. |
| D1 & D2 | Values of the individual components (e.g., cost or concentration). |
| Replacement | Removing a portion of a mixture and replacing it with another component. |
| Replacement Formula | Final = Initial × (1 – Removed/Total)ⁿ, where n = number of times replaced. |
| Cross Difference Trick | Used in alligation: (Higher – Mean) : (Mean – Lower) forms the mixing ratio. |
| Successive Replacement | Used when the process of replacement happens more than once. |
What are the Different Types of Mixtures in Alligation?
When two or more substances are combined, they form a mixture. Alligation is a quick mathematical method to solve problems related to mixtures and their proportions. It helps in finding either the mean value of a mixture or the ratio in which items should be combined.
Mixture
Mixtures can be of two types:
1. Simple Mixtures
These are created by directly combining two or more different items. They are the most basic type of mixtures and are often easy to understand.
Example: An alloy like bronze (made from copper and tin) or steel (made from iron and carbon) is a simple mixture.
2. Complex Mixtures
A complex mixture is formed when the substances being mixed are themselves already mixtures. Solving problems on these requires more steps because you deal with multiple ratios.
Example: If you mix two different alloys (each made from different metals), you get a complex mixture.
Alligation
There are two main types of alligation:
1. Alligation Medial
This is used when the quantity and value (price, percentage, or concentration) of the ingredients are known. Here, you calculate the average or overall value of the mixture.
It works just like a weighted average.
Example: If you mix milk worth ₹40 per litre with water worth ₹0, the average price of the mixture will lie between ₹0 and ₹40.
2. Alligation Alternate
This method is used when you already know the final value of the mixture and want to find the ratio in which two ingredients should be mixed.
It is slightly more advanced but very useful in creating mixtures of a desired value.
What Are the Types of Mixture and Alligation Questions in Quantitative Aptitude?
There are various types of Mixture and Alligation questions that are commonly asked in examinations. The details are as follows:
- Direct Mixture Questions: Simple average or weighted mean calculation
- Allegation Rule Application: Find ratio using cross-difference
- Replacement Type: Quantity removed and replaced repeatedly
- Mixed Concept: Mixture + profit-loss or percentage-based puzzles
Mixture and Alligation Formulas for Quantitative Aptitude
The details of the formulas used to solve Mixture and Alligation questions are as follows:
Properties of Mixture and Alligation
When substances are mixed, whether liquids or solids, some fixed rules always apply. These rules help in solving mixture problems in maths and in real life.
| Property | Details | Example |
| 1. Mean Value or Mean Price | The mean value is the average cost, concentration, or percentage of the mixture. It always lies between the values of the two ingredients. | If rice costs ₹60/kg and wheat costs ₹40/kg, then the mixture price will lie between ₹40 and ₹60. |
| 2. Rule of Alligation | This rule helps to find the ratio in which two items should be mixed to obtain a mixture of a required value. | If tea costs ₹300/kg and another tea costs ₹200/kg, and the average value required is ₹250/kg: Ratio = (250–200):(300–250) = 50:50 = 1:1. |
| 3. Mean Value Always Lies Between the Two Costs | The value of a mixture will never be lower than the cheaper item or higher than the costlier item. | If petrol costs ₹110/litre and kerosene costs ₹80/litre, the mixture price will always lie between ₹80 and ₹110. |
| 4. Proportional Relation of Quantity and Value | The closer the mean value is to one item, the more of that item is used in the mixture. | If the average cost is closer to kerosene (₹80/litre) than to petrol (₹110/litre), then more kerosene will be used in the mixture. |
| 5. Application in Percentages and Concentrations | The same rule applies when working with percentages or concentrations. | Mixing a 10% salt solution with a 40% salt solution will give a concentration between 10% and 40%. |
| 6. Additive Value of Mixtures | The total value of the mixture equals the sum of values of all ingredients. | If 3 kg of rice (₹50/kg) and 2 kg of wheat (₹40/kg) are mixed, total value = (3 × 50) + (2 × 40) = ₹150 + ₹80 = ₹230. |
| 7. Replacement Property | When part of a mixture is removed and replaced with another substance, the concentration/value changes but can still be calculated by alligation. | If 2 litres of a 30% sugar solution are taken out from a 10-litre container and replaced with water, the final concentration reduces and can be found using alligation. |
Mixture and Alligation Tricks for SSC CGL and Other Exams
Some of the solved questions from the Mixture and Alligation topic are as follows:
- Use Alligation Diagram to reduce steps in ratio problems.
- Memorize cross-difference trick to apply allegation instantly.
- In replacement problems, directly apply geometric formula.
- Use weighted average only when quantities differ significantly.
- For profit-based mixtures, set up mean as SP and inputs as CPs.
- Mark options & back-solve when allegation is confusing.
Solved Mixture and Alligation Questions from 2024–25 Exams
Some of the solved questions from Mixture and Alligation topic are as follows:
Asked in SSC CGL 2024 Tier 1 Shift 2 – Memory-Based
Q: A mixture contains milk and water in the ratio 3:1. 20 liters of the mixture is replaced by water. New ratio becomes 1:1. Find initial quantity.
Answer: 40 liters
Explanation: Let total = 40 → milk = 30, water = 10 → remove 20 liters → milk left = 15 → add 20 water → new ratio = 15:25 = 3:5 → Not 1:1
Try 60 → works.
From IBPS PO Prelims 2024
Q: In what ratio should tea worth ₹30/kg be mixed with tea worth ₹50/kg so that the mixture is worth ₹36/kg?
Answer: 7:1
Explanation:
Use alligation:
Based on memory – RRB Group D 2025
Q: A container has 40L milk. 10L is removed and replaced with water. This is repeated once more. Final quantity of milk?
Answer: 24.75L
Explanation:
Use replacement formula:
Mixture and Allegation Concepts for Bank Exams
In IBPS and SBI, the focus is often on:
- Coded Allegation: prices or concentrations are represented by symbols.
- Mixture + Profit: Calculate gain on spurious mixtures.
- Successive Replacement: Repeated removal and addition of components.
Example:
Common Mistakes to Avoid while Solving Mixture and Alligation
Most aspirants make common mistakes while solving questions from this topic. You must avoid the following mistakes while solving Mixture and Alligation questions:
- Using simple average instead of weighted mean.
- Misapplying the allegation rule with wrong order.
- Forgetting the replacement formula’s exponent.
- Not converting percentages into proper ratios.
- Mixing quantity units (litres vs kgs) without checking context.
What Are Related Topics I Should Revise Next?
Topics you should revise next are as follows:
- Questions based on Time and Distance
- Rules and Practice Questions of Simplification
- Shortcuts and Patterns of Number Series
- Various types of Questions Based on Data Interpretation
- Tricks & Examples of Inequalities
FAQs
Yes. In Banking Mains exams, especially IBPS PO or SBI PO, the concept is embedded within DI charts, pie graphs, or caselets involving milk-water or price-mixture-based sets.
The basic formula is:
Quantity of B/Quantity of A=M−D1 / D2−M
Use alligation when you know the values and mean, and need to find the ratio or quantity of components to mix.
Yes, but it becomes complex. Use pairwise or weighted average methods instead for three or more items.
Use the weighted average formula, then apply alligation to derive the ratio for unequal quantities.
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